Solutions and Improved Perturbation Analysis for the Matrix Equation X-A*X-pA=Q   (p>0)

Abstract

The nonlinear matrix equation X-A*X-pA=Q with p>0 is investigated. We consider two cases of this equation: the case p≥1 and the case 0<p<1. In the case p≥1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case 0<p<1, a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.